By [1]}, two halfrays of \(A_i\)  and \(A_j\)  emanating from \(x\)  are contained in an apartment. Thus the projection of \(A_j\)  onto \(A_i\)  is equal to their intersection point.
By our assumption on the intersection points, and the fact that projection does not increase distances [2]}, the proof is completed using Theorem REF .
